Animate collection instance (material) independently?
Sure.  Check it out:



There are a number of different ways to identify exactly which emitters you want behaving this way or that, but I think that using UV is the easiest, particularly with 2.8 improvements.  Here, I'm using the UV.x position of each object, which has been shrunk to 0 with individual origins pivot, to decide what period to use for the sine wave that I'm using to drive emissions.  Everything is driven from a single value node, where I just entered #frame to get my current frame count.

You can use something other than UV to identify which objects you want behaving which way, like object ID or position or vertex color.  I just find UV easy to use and flexible.  If you're using UV already, just create a new UV map; but if you're using linked objects, you'll need to use a different identifier.